The decision and its conditions
A graph transformation acts on input and output coordinates differently. For g(x)=f(b(x−h))+k, a known input u of f corresponds to x=h+u/b, while its output becomes f(u)+k. A reflection may change the input, the output or both; read the formula rather than copying a direction from memory. Even functions satisfy f(−x)=f(x), and odd functions satisfy f(−x)=−f(x) on a symmetric domain. An exponential graph can approach a horizontal line without reaching it at any finite input.
Worked example
The point (4,6) lies on y=f(x). For g(x)=f(2(x−3))+3, what x-coordinate corresponds to this point? The inner input must equal 4: 2(x−3)=4. Solve x=3+4/2. The corresponding x-coordinate is 5; its output is 9.
A contrasting decision
The table gives f. If g(x)=−f(−x)+4, find g(3). Figure data: Values of f. Columns Input, Output. Rows: (-3, 6), (0, 2), (3, 10). First reverse the input to −3; the table gives f(−3)=6. Then reverse that output sign and add four. g(3)=−6+4=-2.
Diagnose a competing method
The first example’s tempting answer 11 follows this mistake: Multiply the old input by two instead of dividing. Solve for the new input; a factor inside f acts reciprocally on horizontal coordinates.
Transfer before revealing
For f(x)=x³+(k+5)x²−2x to be odd for all real x, what must k be? Pause and identify the original restrictions. Solution: An odd polynomial has no nonzero even-power terms. The coefficient of x² must vanish: k+5=0. Therefore k=-5; the cubic and linear terms already have odd powers.
Check the boundary of the method
Even functions satisfy f(−x)=f(x), and odd functions satisfy f(−x)=−f(x) on a symmetric domain. The two negative signs act at different stages: one on input, one on output. Check the solution in the original statement. A transformed equation can generate candidates rather than equivalent solutions.
The point (4,6) lies on y=f(x). For g(x)=f(2(x−3))+3, what x-coordinate corresponds to this point?
- The inner input must equal 4: 2(x−3)=4.
- Solve x=3+4/2.
- The corresponding x-coordinate is 5; its output is 9.
Compare the other answer choices
Ignore the horizontal scale factor.
Report the transformed output coordinate instead of x.
Multiply the old input by two instead of dividing.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
CLEP College Algebra exam overview